Post on 31-May-2020
1
MODEL VALIDATION FOR THE OPTIMIZATION OF REFINERY PREHEAT TRAINS
UNDER FOULING
*F. Lozano-Santamaria1, and S. Macchietto1 1 Department of Chemical Engineering, Imperial College London, South Kensington Campus, London, SW7
2AZ, UK (fl3015@ic.ac.uk)
ABSTRACT
Fouling in refinery (and other) applications is a
major issue affecting process efficiency and
profitability. Many models are available that aim to
quantify its effects on individual exchangers and
whole networks, and the benefits of fouling
mitigation techniques. A key question is how much
model detail is required versus intended use,
predictive accuracy, data requirement, and
computational feasibility. This paper compares two
dynamic thermo-hydraulic models for heat
exchanger networks under fouling: a high fidelity
model (A) suitable for simulations, and a simpler,
model (B) suitable for optimisation. It identifies
conditions and applications where the two models
are broadly equivalent, presents a parameter
estimation scheme to match them, and a validation
methodology. A detailed comparison between the
models is made for 37 exchangers in 8 networks.
Results show that the simpler model (with
parameters suitably fitted as indicated) can
approximate well the high fidelity model for
relatively long periods. The simpler model B is then
successfully used to simultaneously optimize
cleaning schedule and flow distribution of a pre heat
train. This solution is validated against model A,
with a difference in predicted operational cost of
1.4% over 1 year. Results indicate that the simpler
model and fitting procedure approximate closely the
high fidelity model over relatively long periods, and
can be confidently used in a nonlinear model
predictive control (NMPC) strategy.
INTRODUCTION
Fouling in the preheat train of crude distillation
units (CDU) reduces the thermal and hydraulic
performance of the process, and increases
operational cost and environmental impact.
Choosing the right mitigation technique is
challenging due to many possible alternatives,
complex trade-offs , and difficulty in quantifying
future performance and benefits [1]. Suitable
models, able to predict and quantify the effects of
fouling in the preheat train, are therefore essential.
Operational mitigation techniques (cleaning the
units and control of flow distribution in the network)
improve operations without major design
intervention and large capital investment.
Mathematically, this requires solving dynamic
optimization problems with integer variables, to
minimize the total operating cost of the network
over a specified horizon [1], [2]. Computationally,
these are very demanding problems. Clearly, the
effectiveness of a solution depends on the accuracy
of the underlying models, many of which have been
proposed. Some use linearised heat exchanger
models [3], [4], others consider simple fouling
models (e.g. linear fouling, constant fouling) [5]โ
[7]. Details of exchanger geometry, fluid and deposit
thermo-physical properties, and interactions
between units in the network are often ignored or
highly approximated. Such simplifications make the
optimization problem computationally easier to
solve, but compromise the quality of the results.
With more detailed thermo-hydraulic models, the
optimization is compromised, having to rely on
heuristic or stochastic methods with poor ability to
deal with constraints and no guarantee of
convergence to an optimum point (e.g. [8]).
A key question is how much model detail is
required for an intended use, in terms of predictive
accuracy, data requirement and computational
feasibility. Intended uses may be exchanger and
exchanger network design and, in operations,
monitoring of fouling extent, diagnosis of abnormal
events such as inorganics breakthrough, flow rate
control, and planning and scheduling of cleanings.
So far, most of these have been addressed using
distinct models, ranging from very simple, Rf-based
to CFD models (for a good discussion, see [9]). A
trade-off is clearly necessary between tractable
models suitable for optimization, and more accurate
models which are too large and complex for this.
In this paper, the intended model use is the
optimization of flow control and cleaning schedules,
which have been shown to be highly synergistic
[10]. Two thermo-hydraulic heat exchanger models
are compared: Model A, a high fidelity, 2D-
Heat Exchanger Fouling and Cleaning โ 2019
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2
distributed (axially and radially) dynamic model,
and model B, a much simpler 1D, radially
distributed but axially lumped dynamic model. The
aim is to determine whether, to which extent and
under which conditions model B can approximate
model A in predicting the effects of fouling. An
application of approximate but tractable models is
online optimization within a nonlinear model
predictive control strategy for fouling mitigation.
In the following, first some details of the two
models (A and B) are given, with a comparison of
their key features. Then, a scheme is presented for
fitting the simpler model B to the high fidelity model
A, which is taken as reference. The errors achieved
in the estimation step, and in a subsequent prediction
step using the simpler model are analyzed for 37
exchangers in 8 networks. Finally, a network
optimization is carried out with model B and the
results checked against the high fidelity model,
showing that a very good approximation is achieved.
Key conclusions are summarized in the last section.
HEAT EXCHANGER, NETWORK AND
FOULING RATE MODELS
Although the following approach is general,
only tube and shell heat exchangers are considered
here, due to their large presence in the oil industry.
A preheat train model must include individual
exchangers and how they are connected in a
network. A heat exchanger network is defined as a
multigraph that describes the nodes, the connections
between them, and the direction of flow among the
nodes for different streams. Nodes are classified as
sources (๐๐), sinks (๐๐), flow splitters (๐๐), flow
mixers (๐๐ฅ), and exchangers (๐ป๐ธ๐). Each stream
defines a unique graph which connects some nodes.
Only in the exchanger nodes two streams interact via
heat transfer. Fig 1 shows a network example
defining all the nodes, and a furnace. The latter
performance is modelled simply, by defining the
coil inlet temperature (CIT), the coil outlet
temperature (COT), throughput, and furnace duty. A
network is defined by the following sets:
โข ๐๐๐๐๐ = ๐ป๐ธ๐ โช ๐๐ โช ๐๐ฅ โช ๐๐ โช ๐๐. Set
of all the nodes in the network.
โข ๐๐ก๐๐๐๐ ๐ก๐ฆ๐๐๐ = {1,2, โฆ , ๐๐ ๐ก}. Set of all
the fluids in the network.
โข ๐ด๐๐๐ = {(๐, ๐, ๐)|โ(๐, ๐, ๐) โ ๐๐๐๐๐ ร๐๐๐๐๐ ร ๐๐ก๐๐๐๐ ๐ก๐ฆ๐๐๐ }. Set of arcs that
defines the connection between nodes for a
given fluid.
โข ๐๐๐๐ = {1,2, โฆ , ๐๐}. Set of discrete points
in time.
Each element in the set ๐๐ก๐๐๐๐ ๐ก๐ฆ๐๐๐ is a fluid
in the network (e.g. crude oil, naphtha). The set ๐ด๐๐
represents a connection between two nodes
including the fluid that connects them. Finally, the
set ๐๐๐๐ is not a physical entity, but is included
because dynamic processes may be defined
differently in different solution algorithms. The
source nodes define the inlet flow rates and inlet
temperature of a specific stream at every time. Each
heat exchanger node is defined by the corresponding
heat and mass transfer, fouling and deposit models
(depending on model type). Here, network
configuration and exchangers design are fixed.
Fig 1. Heat exchanger network representation
Given all inputs to the network (inlet streams
temperature, flowrate and enthalpy), stream physical
properties models (e.g. for density, heat capacity),
and exchanger geometries, the aim is to predict the
performance (in term of outlet streams temperature,
duty and pressure drop in each node and for the
network overall) under varying operating
conditions, including fouling.
The models for an individual heat exchanger
and fouling are discussed in more detail in the
following, but the network model is invariant for any
changes in the fouling or exchanger models.
Fouling rate model
Crude oil fouling is a complex process in which
many fouling mechanisms take place [11]. A semi-
empirical approach has traditionally been favored
[12] because these models are easy to implement
and understand, and are claimed to capture the main
effects of operating variables on the fouling rate.
Most semi-empirical models in the literature state
that the fouling rate is a function of two competing
phenomena: a deposition rate, and a removal (or
suppression) rate. Under certain conditions these
balance out, defining a fouling threshold level.
Fouling rate is also traditionally (and incorrectly, see
[13]) equated with fouling resistance. For example,
the Ebert-Panchal (EP) model, in the form of Eq. 1,
has been widely used in crude oil applications [12],
[14]. It defines fouling rate as a function of the
Prandtl and Reynold number (reflecting the effects
of fluid properties and operating conditions of the
exchanger), and an Arrhenius term (reflecting the
chemical reaction nature of crude oil fouling) [15],
[16]. The removal (suppression) term is proportional
to the shear rate, hence velocity in the tubes.
๐๐ ๐
๐๐ก= ๐ผ๐ ๐โ0.66๐๐โ0.33 exp (โ
๐ธ๐
๐ ๐๐
) โ ๐พ๐ (1)
By assuming constant thermal conductivity and
uniform composition of the deposit, its thickness
HEX2
Crude
Naphtha
HEX1
Mx1 Sp1
Mx2
Sp2
So1
So2
Si2
Si1
Furnace
CIT COT
Mixers
Sources
Splitters
Sinks
Exchangers
Stream types
Heat Exchanger Fouling and Cleaning โ 2019
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(hence the hydraulic performance of an exchanger)
can be calculated. The EP model has three tuning
parameters: the deposition constant, ๐ผ, the removal
constant, ๐พ, and the activation energy, ๐ธ๐, which
must be determined from plant or experimental data.
Assuming that the same set of parameters apply to
many exchangers in the same network simplifies the
parameter estimation problem by reducing the
number of variables [5], [17]. However, there is no
reason to assume this. Allowing different fouling
parameter for each exchanger in the network can
improve the overall prediction capabilities of the
model, although it means solving a parameter
estimation problem with more variables.
Here, we use the Ebert-Panchal model to
represent deposition rate locally (model A) or, in the
form of (Eq. 1), overall fouling resistance (model B).
Heat exchanger model
Given as inputs: the flow rates of the tube and
shell side, the temperature of the hot and cold
streams, the physical properties of the fluids, and the
geometry specification of the exchangers, a heat
exchanger model (coupled with a fouling model)
must be able to predict: i) the outlet stream
temperature of the fluids on the tube and shell side;
ii) the heat transferred from the hot stream to the
cold stream and iii) the pressure drops on the tube
and/or the shell side.
Here we focus on two models: (A) a high
fidelity, 2D dynamic model, and (B) a much simpler
1D dynamic model. Both models use the same input
information described above and are able to predict
the main operating variables of the exchanger. The
key difference is in the level of detail with which
they account for the geometry of the exchanger. This
has significant implications on model complexity,
ease of solution, and prediction capabilities.
Model A: this 2D distributed dynamic model
[18], [19] includes a detailed description of the
energy balances and hydraulic effects. These are
defined by a set of algebraic and partial differential
equations along both the axial and radial directions
(hence, 2D distributed), for shell and tube
exchangers. Differential equations are discretized
and solved using a state of the art numerical
integrator. Details are found in [18], [19].
This model considers four domains (tube side
per pass, shell side, tube wall, and deposit), linked
by boundary and continuity conditions. A tube side
domain is associated to each pass of the exchanger,
with a single tube representing the whole bundle. On
the shell side domain, the heat flux from all passes
is included in the differential energy balance. In the
wall and deposit domains the energy balance is
solved in the radial direction. The deposit domain
has a moving boundary layer (handled using a
Lagrangean transformation [18], [19]), as the
thickness of the deposit changes with time and with
the axial position in the exchanger. Each point in the
deposit (radially and axially) is characterized by its
own thermal conductivity, which changes over time
reflecting operation history. This high fidelity model
has demonstrated excellent ability to predict plant
measurements and performance in many
applications, for single exchangers and large
networks. It has been validated against refinery data
[18], [20], [21], and used for monitoring, diagnostic
and retrofit of industrial networks [9], [22], [23].
Simulations are performed in reasonable
computational time, however its large size hinders
its application to optimisation.
Model B: this simpler radially-distributed but
axially-lumped parameter exchanger model (hence,
1D-distributed) considers the overall effect of the
exchanger inputs on the outputs, without much
detail for the heat transfer inside the unit. It is based
on the P-NTU model [24] and the definition of the P
efficiency (the actual heat transfer in the exchanger
with respect to the maximum possible heat transfer),
which is related to the number of transfer units and
the exchanger geometry. The time evolution of the
system is given by the fouling/agein model, and at
every time a steady state algebraic model for the
exchangers determines the outlet streams
temperature and pressure.
Model B considers the deposit thickness (hence
pressure drop) based on the fouling resistance,
accounting for curvature effects on heat transfer in
the radial direction (Eq. 2). Notably, Eq. 2
overcomes the usual thin layer assumption of similar
models (e.g. [25]).
๐ฟ =๐๐
2[1 โ exp (โ
๐๐๐ ๐
๐๐ 2โ)]
(2)
Although models (A) and (B) have similar
inputs and can predict the same outputs for each
exchanger, their level of detail and mathematical
complexity are significantly different, and their
applicability may also be different. Table 1 shows a
qualitative comparison of the two models in terms
of number of equations after discretization. For both
models, the time domain is discretized using
orthogonal collocation in finite elements [26].
Model A uses finite differences in others domains.
The intended application here is the
optimization of flow rate distribution and cleaning
schedule over long horizons. For flow control, the
decision variables (flowrates) are continuous. For
the cleaning scheduling it is necessary to introduce
binary decision variables (to clean or not) at each
time of interest, for each exchanger. The number of
binary variables increases rapidly with problem size.
They are the hardest to tackle in an optimization
problem, and should be reduced to a minimum. For
the optimal cleaning scheduling formulation, they
also introduce disjunctions (logical OR, e.g. a unit is
either in operation or being cleaned). Their
complexity is directly related to the size and
Heat Exchanger Fouling and Cleaning โ 2019
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characteristics of the model. A large model means a
more difficult optimization problem to solve (a
comprehensive complexity analysis is given in
[27]). From Table 1 the size of the high fidelity
model A can be well over ten times that of model B.
It has therefore a very large disadvantage with
respect to model B.
Table 1. Model comparison and estimation of the number of equations. Lumped parameter model Distributed model
Algebraic (A)
or differential
(D) equation.
Estimated number of
equations after
discretization*.
Algebraic (A) or
differential (D)
equation.
Estimated number of
equations after
discretization*.
Mass balance A (2*NHEX)*NT A (2*NHEX)*NT
Energy balance A (2*NHEX)*NT D (NHEX*NZ*NP*(2*NR+1))*NT
Fouling model D (NHEX)*NT D (NHEX*NZ*NP *NR)*NT
Deposit thickness A (NHEX)*NT D (NHEX*NZ*NP)*NT
Heat transfer A (3*NHEX)*NT A (2*NHEX*NZ*NP)*NT
Pressure drop A (NHEX)*NT D (NHEX*NZ*NP)*NT
Total (10*NHEX)*NT NHEX*(2+NZ*NP*(5+2*NR))*NT
*NHEX, number of heat exchangers in the networks (โฅ 1).
NT, number of discretization points in time (โฅ 2).
NZ, number of discretization points in the axial direction (โฅ 5).
NR, number of discretization points in the radial direction (โฅ 20).
NP, number of tube passes (โฅ 1).
PARAMETER ESTIMATION AND MODEL
VALIDATION
The prediction ability and accuracy of the
models are compared for a collection of heat
exchanger networks operating in different
conditions. Model A is used as a reference
benchmark as it was validated against many plant
data, with reported error of +/- 2.0 K in outlet
streams temperatures, and duty and tube side
pressure drop within a 1.5% relative error [9], [18].
Fig 2. Model validation strategy.
The model validation process adopted,
summarized in Fig 2, starts with the generation of
operational data for each case study by running one
or more scenarios with the high-fidelity model A.
Model A had been previously fitted (i.e determining
fouling parameters) and validated against real
dynamic plant data collected from the refinery. Then
simulated data for the same inputs are generated
from model A (including soft-measured variables)
and divided in two subsets: data in the estimation
horizon (EH) are used for fitting parameters (๐ผ, ๐ธ๐.
and ๐พ in the EP model, plus deposit roughness) in
model B, in a network-wide estimation; those in the
prediction horizon (PH) are used for validating the
prediction of model B against those of model A. The
length of the estimation horizon (EH) may be varied.
Fig 3 shows that model B uses a full set of
temperature and pressure โmeasurementsโ produced
using model A. In practice, this would include some
pressures made available through model A as soft-
sensed variables. The fitting also considers the effect
of complex interactions between units on the
network.
๐ฆ๐ข๐ง ๐ฑ = โ โ ๐๐ป๐(๐ป๐,๐,๐ โ ๏ฟฝฬ๏ฟฝ๐,๐,๐)๐
๐โ๐ฏ๐ฌ๐ฟ๐โ๐ต
+ ๐๐ป๐(๐ป๐,๐,๐ โ ๏ฟฝฬ๏ฟฝ๐,๐,๐)๐
+ ๐๐ซ๐ท(๐ซ๐ท๐,๐ โ ๐ซ๏ฟฝฬ๏ฟฝ๐,๐)๐
๐. ๐. ๐ฏ๐๐๐ ๐๐๐๐๐๐๐๐๐ ๐๐๐ ๐๐
๐น๐๐ข๐๐๐๐ ๐๐๐๐๐ ๐๐๐ก๐ค๐๐๐ ๐๐๐๐๐๐๐ก๐๐ฃ๐๐ก๐ฆ ๐๐๐ ๐ ๐๐๐ ๐๐๐๐๐๐ฆ ๐๐๐๐๐๐๐๐ ๐๐๐๐ ๐ ๐ข๐๐ ๐๐๐๐
๐๐๐๐๐๐ก๐๐๐๐๐ ๐๐๐๐ ๐ก๐๐๐๐๐ก๐
(3)
The parameter estimation problem solved here
(Eq. 3) minimises the square error of the difference
in measurements (tube side outlet temperature, shell
side outlet temperature,and tube side pressure drop)
between models A and B. It determines optimally
the fouling parameters and deposit roughness model
B. The weights ๐ค define the relative importance of
Dynamic plant
data
Fitting distributed
model (A)
Network simulation with A
and data collection
Dynamic
simulated data for
fitting
Dynamic
simulated data for
validation
Fitting lumped
model (B)
Is model (B)
valid?
โข Network
troubleshooting.
โข Multi component
deposition detection.
โข Operational diagnose.
โข Network performance
analysis.
โข Optimal control.
โข Optimal cleaning
scheduling.
โข Simultaneous fouling
mitigation strategies.
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each objective. The optimization problem is subject
to all equations of the model, including network
connectivity, pressure drop, operational constraints
(e.g. bounds on variables) and heat
exchanger/fouling models, which are nonlinear
equations. The size of this NLP optimization
problem depends on the size of the network and on
the number of available data points (i.e. length of
EH) and sampling frequency. This estimation
procedure is therefore quite different from, much
richer than, and with superior results to, those
typically used for fitting fouling parameters. In
approaches that rely on the calculation of the fouling
resistance (๐ ๐) (boxes at the bottom of Fig. 3) the
noise and error of the primary measurements are
amplified [5], [28] and only temperature
measurements are used. Alternative methods
(dashed line in Fig.3) cannot exploit the full set of
pressure drop information made available by the soft
sensors in model A [9]. In addition, here is possible
to assess the quality of model B estimates (and
predictions) against the high fidelity model.
Fig 3. Model fitting and validation approached, from plant data to a representative model.
This methodology is applied to eight networks
taken from the literature, all at the hot end of preheat
trains [9], [21], [22], [27]. The networks range from
small (2 exchangers) to relative large ones with 9
exchangers. The total of 37 heat exchangers cover a
wide range of operating conditions and design
specifications and provide a significant sample. In
all the cases the overall operation time is set at 365
days. Initial conditions are case dependent and some
exchangers present an initial fouling resistance. The
inlet conditions in all the cases are assumed constant
(no changes in the flow rate or inlet temperatures).
All cases are solved for the following EH: 360 days,
270 days, 180 days, and 90 days (the PH is the
balance to the end of the year). This allows checking
the minimum information required to fit a model and
prediction quality.
Hexxcell studio [29] is used to simulate the
operation of the networks using distributed model A
and generate the data for fitting model B. The results
are exported to Python, and the parameter estimation
problem is solved in Pyomo 5.2 [30] using the solver
IPOPT, an interior point algorithm for NLP
problems.
RESULTS AND DISCUSSION
The results are divided in two groups. The first
one presents the results of the parameter estimation
problem, and compares the two models for all
exchangers studied. The second one optimizes the
operation a specific case study with model B, and
validates the solution obtained against model A.
Parameter estimation and model comparison
The average absolute error between models A
and B is calculated for the primary measured
variables: outlet tube side temperature, outlet shell
side temperature, and tube side pressure drop. Fig 4
plots the errors for the estimation data set (EH error,
for data within the model fitting horizon) and
prediction data set (PH error, for data not included
in the model fitting) for all 37 exchangers, for
various estimation horizons, EH. The average
estimation error between the two models is low
(โค1.5 K for the tube side temperature, โค 2.0 K for
the shell side temperature, and โค 0.15 bar for the
pressure drop) and does not change significantly
with the length of the estimation horizon. This
indicates that the simpler model B is able to match
well the high fidelity model A. The average
prediction error is larger, particularly for short
estimation horizons, however still quite acceptable.
In isolated cases, an estimation horizon of 90 days
leads to >5 K prediction errors in the shell side outlet
temperature. Including too few data in the parameter
estimation step may lead to wrong long term
Dynamic plant data
(Temperature, pressure, flow rates)
(T, P, F)
Data filteringFitting 2D distributed
model
Simulation of the
network
Fitting 1D lumped model
Model
validation
Calculate fouling
resistance (Rf) for each
HEX
Fitting fouling model
(e.g. Ebert-Panchal)
Alternative and
simplified lumped model
Evaluate performance of the
network and optimize its operation
Th
is p
aper
ap
pro
ach
Oth
er a
ppro
ach
es
Direct
approach
Only temperature
measurements
Include thermal and
hydraulic effects
Indirect
approach
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predictions. In some case the large variation in
prediction error is due to bad parameter estimates for
the B model. In others, the estimation data are not
sufficient to capture the network variability.
Estimation horizons of ~180 days give good results.
a)
b)
c)
Fig 4. Average error for a) tube side outlet
temperature, b) shell side outlet temperature, and c)
tube side pressure drop in the estimation horizon
(EH) and in the prediction horizon (PH).
Fig 5 shows values of the estimated deposition
constant parameter (๐ผ) in each exchanger vs the
estimation horizon. In some cases, as the estimation
horizon decreases the value of ๐ผ increases (e.g.
exchangers no. 10-15 in Fig 5). Although they give
a good fit within the estimation horizon, these large
values overestimate the effect of fouling for longer
operating times causing erroneous predictions. In
model A the same fouling parameters were used for
all the exchangers in a network, in the simpler model
B different values are estimated for each exchanger.
This additional degrees of freedom enables model B
to make equally good predictions when enough data
is used to fit the parameters of the simpler model.
Fig 5. Optimally determined deposition constant
(๐ผ) for each exchanger of all the cases studied
varying the estimation horizon.
Network optimization and validation
The heat exchanger network of Fig 6 (taken
from [9]) is considered here. After fitting the
parameters as described above, model B is used to
simultaneously optimize its cleaning schedule and
flow distribution, over 365 days. Here, the objective
function is the minimization of the total operating
cost (fuel cost + CO2 emission cost + cleaning cost).
Details of formulation and solution strategy are in
[10], [27]. The optimal cleaning schedule for this
network consist on cleaning E01A/B and E04 four
times. In three occasions those three shells are
cleaned simultaneously (at 30 days, 200 days, and
290 days of operation). The other cleanings are at
110 days for E4 and 120 days for E01A/B. The
optimal flow split through the parallel branches,
bounded between 30% and 70% for feasible
operation, changes during the year (Fig 7) to
enhance energy recovery when the units
performance changes due to fouling and/or the
cleanings. The optimal operation is then run with
model A and the trajectories from the two models
are compared. Fig 8 shows the CIT prediction for
this optimal operation for the simpler model B, and
the corresponding profiles when the same operation
is run with the high fidelity model A. Both models
follow very similar trends. The average error in CIT
prediction between the two over 1 year is 0.31 K.
Model B predicts an overall operation cost of $
13.7 M, while model A predicts $ 13.4 M, a
difference of only 1.4%. Therefore, the operational
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decisions, including choice of cleanings and flow
distribution, made with the model B are taken to be
valid
Fig 6. Heat exchanger network structure used as an
example in comparing the models (A) and (B).
Fig 7. Optimal split fraction for the case study.
Fig 8. Comparison of the CIT profile for the case
study using the simple and high fidelity models.
CONCLUSION
Fouling mitigation in refinery applications is
paramount to ensure a profitable, safe, and reliable
operation. Accurate models are necessary to predict
the network performance under fouling and reliable
predictions to support fouling mitigation decisions.
A detailed comparison of two dynamic heat
exchanger models (a high fidelity, 2D distributed vs
simpler 1D distributed) was presented, as well as a
methodology to optimally estimate the parameters
of the simpler model, without relying on indirect
quantities such as the calculated fouling resistance.
This parameter estimation approach considers all the
interactions among the units in the network and
exploits all measurements, including any soft-
sensed pressure drop, and incorporates them into
individual tuning parameters for each exchanger.
The proposed parameter estimation
methodology was applied in 8 heat exchanger
networks with a total of 37 exchangers. With an
estimation horizon of appropriate length, the simpler
model approximates the predicted performance of
the network within a small error, relative to the high
fidelity model. The much smaller size of the simpler
model enables its use in advanced optimal fouling
mitigation formulations, in particular for the
demanding simultaneous optimal cleaning
scheduling and flow distribution control problem,
which was shown to be highly beneficial [10], [27].
This was confirmed here with a case study, where
the combined optimal strategy generates significant
savings. The optimal operation thus calculated
(cleaning schedule and flow split control profiles
over 1 year) was validated against the responses
obtained with the full high fidelity model. The error
between the models in the predicted operational cost
is 1.4%. This confirms that the simpler model (with
the proposed parameters fitting procedure) can
approximate closely the more complex model over
relatively long periods. It also indicates it can be
confidently used within a nonlinear model
predictive control (NMPC) strategy.
NOMENCLATURE
๐๐ Tube inner diameter, mm
๐๐ Tube outer diameter, m
๐ธ๐ Fouling activation energy, kJ/kmol
๐๐ Prandtl number
๐ Universal gas constant, J/molK
๐ ๐ Reynolds number
๐ ๐ Fouling resistance, m2K/W.
๐๐ Film temperature, K
๐๐ก Tube side outlet temperature, K
๐๐ Shell side outlet temperature, K
๐ Pressure, bar
๐ค Objective weight
๐ผ Deposition constant, m2K/W day
๐พ Removal constant, m4K/N W day
๐ฟ Deposit thickness, m
๐๐ Deposit thermal conductivity, W/m K
๐ Shear stress, N/m2
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